算額（その93）

算額（その93）

千葉県君津市 神野寺 明治20(1887)年4月
http://www.wasan.jp/chiba/jinyaji2.html

甲円の下に乙円，丙円，丁円がある。各円の径を求めよ。

甲円の半径を 1 として，以下のように記号を定め，方程式を解く。
なお，solve() では解が求まらないので，nlsolve() を使用し，数値的に解を求める。

using SymPy
@syms r1::positive, r2::positive, r3::positive, x1::positive, x2::positive, x3::positive, x::positive, y::positive;
r = 1
x1 = r1 
eq1 = x*x3 + y*(r3 - y) |> expand
eq2 = r3*x - y*(x - x3) |> expand
eq3 = x1^2 + (y - r1)^2 - (r + r1)^2 |> expand
eq4 = x2^2 + (y - r2)^2 - (r + r2)^2 |> expand
eq5 = x3^2 + (y - r3)^2 - (r + r3)^2 |> expand
eq6 = (x2 - x1)^2 + (r2 - r1)^2 - (r2 + r1)^2 |> expand
eq7 = (x3 - x2)^2 + (r3 - r2)^2 - (r3 + r2)^2 |> expand;

# solve([eq1, eq2, eq3, eq4, eq5, eq6, eq7], (r1, r2, r3, x2, x3, x, y))

eq1 |> println
eq2 |> println
eq3 |> println
eq4 |> println
eq5 |> println
eq6 |> println
eq7 |> println

   r3*y + x*x3 - y^2
   r3*x - x*y + x3*y
   r1^2 - 2*r1*y - 2*r1 + y^2 - 1
   -2*r2*y - 2*r2 + x2^2 + y^2 - 1
   -2*r3*y - 2*r3 + x3^2 + y^2 - 1
   r1^2 - 4*r1*r2 - 2*r1*x2 + x2^2
   -4*r2*r3 + x2^2 - 2*x2*x3 + x3^2

using NLsolve

function nls(func, params...; ini = [0.0])
if typeof(ini) <: Number
r = nlsolve((vout, vin) -> vout[1] = func(vin[1], params...), [ini], ftol=1e-14)
v = r.zero[1]
else
r = nlsolve((vout, vin)->vout .= func(vin, params...), ini, ftol=1e-14)
v = r.zero
end
return v, r.f_converged
end;

function H(u)
(r1, r2, r3, x2, x3, x, y) = u
return [
       r3*y + x*x3 - y^2,
       r3*x - x*y + x3*y,
       r1^2 - 2*r1*y - 2*r1 + y^2 - 1,
       -2*r2*y - 2*r2 + x2^2 + y^2 - 1,
       -2*r3*y - 2*r3 + x3^2 + y^2 - 1,
       r1^2 - 4*r1*r2 - 2*r1*x2 + x2^2,
       -4*r2*r3 + x2^2 - 2*x2*x3 + x3^2
      ];
end;

iniv = [0.1, 0.2, 0.3, 0.4, 0.7, 1.0, 1.1];
res = nls(H, ini=iniv)

   ([0.13839876084731936, 0.17954306798152128, 0.3287706665765487, 0.45366714256276236, 0.9395827489780465, 1.2683534155545952, 1.268353415554595], true)

using Plots

function circle(ox, oy, r, color=:red; beginangle=0, endangle=360)
 θ = beginangle:0.1:endangle
 x = r.*cosd.(θ)
 y = r.*sind.(θ)
 plot!(ox .+ x, oy .+ y, color=color, linewidth=0.5)
end;

function point(x, y, string="", color=:green, position=:left, vertical=:top; fontsize=10, mark=true)
  mark && scatter!([x], [y], color=color, markerstrokewidth=0)
  annotate!(x, y, text(string, fontsize, vertical, position, color))
end;

function segment(x1, y1, x2, y2, color=:black; linestyle=:solid, linewidth=0.5)
plot!([x1, x2], [y1, y2], color=color, linestyle=linestyle, linewidth=linewidth)
end;

function draw(more=false)
   pyplot(size=(500, 500), grid=false, aspectratio=1, label="", fontfamily="IPAMincho")
   (r1, r2, r3, x2, x3, x, y) = [0.13839876084731936, 0.17954306798152128, 0.3287706665765487,
       0.45366714256276236, 0.9395827489780465, 1.2683534155545952, 1.268353415554595]
   r = 1
   plot()
   circle(0, y, 1)
   circle(r1, r1, r1, :green)
   circle(-r1, r1, r1, :green)
   circle(x2, r2, r2, :blue)
   circle(-x2, r2, r2, :blue)
   circle(x3, r3, r3, :magenta)
   circle(-x3, r3, r3, :magenta)
   hline!([0], color=:black, lw=0.5)
   if more
       point(0, y, "y ", :red, :right)
       point(0, y-r, "y-r ", :red, :right)
       point(r1, r1, "(r1,r1)", :green, :top)
       point(x2, r2, "(x2,r2)", :green, :top)
       point(x3, r3, "(x3,r3)", :magenta, :top)
       point(x, 0, "   x", :magenta, :bottom)
       segment(0, y, x, 0, :black)
       vline!([0], color=:black, lw=0.5)
   end
end;